Quantum simulation of chemistry via quantum fast multipole method

TL;DR

An approach for simulating quantum chemistry on quantum computers with significantly lower asymptotic complexity than prior work, using a real-space first-quantised representation of the molecular Hamiltonian which is propagated using high-order product formulae.

Abstract

Here we describe an approach for simulating quantum chemistry on quantum computers with significantly lower asymptotic complexity than prior work. The approach uses a real-space first-quantised representation of the molecular Hamiltonian which we propagate using high-order product formulae. Essential for this low complexity is the use of a technique similar to the fast multipole method for computing the Coulomb operator with $\widetilde{\cal O}(η)$ complexity for a simulation with $η$ particles. We show how to modify this algorithm so that it can be implemented on a quantum computer. We ultimately demonstrate an approach with $t(η^{4/3}N^{1/3} + η^{1/3} N^{2/3} ) (ηNt/ε)^{o(1)}$ gate complexity, where $N$ is the number of grid points, $ε$ is target precision, and $t$ is the duration of time evolution. This is roughly a speedup by ${\cal O}(η)$ over most prior algorithms. We provide lower complexity than all prior work for $N<η^6$ (the regime of practical interest), with only first-quantised interaction-picture simulations providing better performance for $N>η^6$. As with the classical fast multipole method, large numbers $η\gtrsim 10^3$ would be needed to realise this advantage.

Figures (3)

• Figure 1: Illustration of hierarchical division of a simulation cell, as well as the interaction list and neighbours for an exemplary (green) box in the $\ell=4$th level of a tree. Here, the level-1 box is the complete region. The four level-2 boxes are outlined by the red lines. The 16 level-3 boxes are shown by the thick black lines and the level-2 partition. The 64 level-4 boxes are shown by the thin black lines and the level-3 partition. The box of interest $b$ is represented by the green square. The surrounding white squares are neighbours of $b$, and the boxes in $b$'s interaction list are shown in orange. The grey squares are neither nearest neighbours, nor in the interaction list. A two-dimensional simulation cell is presented for simplicity of presentation.
• Figure 2: 1D demonstration of Algorithm 2. Here $\eta=4$, $L=3$, $c=3$, and $n_b = 4$. (a) The positions of the electrons are shown on the real space grid, where, $x_i$ is the position of the $i$th electron. (b) Snapshots of the boxes and position registers throughout Algorithm 2. The label of a given box is shown above and the corresponding registers are shown below. The flag registers are not depicted. Each row iterates through the outer-most loop of Algorithm 2 for $\ell=1,2, 3$. For a given $\ell$, the iteration starts by swapping position data into the correct registers. For $\ell=1$, this step is replaced by an initialisation step. Next, the positions are sorted within each box. This procedure is repeated until $\ell=L$ is reached.
• Figure 3: The interaction list and neighbours used for FMM. The green square is the box $b$ of interest, the white squares are neighbours of $b$, and the boxes in the interaction list are shown in orange. The heavy black lines indicate the boxes for the next level up. The red lines indicate the box two levels up containing the box of interest, with parts (a) to (d) indicating the four alternatives. The blue zig-zag lines indicate the Morton ordering of boxes within that higher-level box.

Theorems & Definitions (2)

• Lemma 1: Morton orderings with shifts
• proof